GENERATION AND PROPAGATION OF INTERFACES IN REACTION-DIFFUSION SYSTEMS · 2018-11-16 · transactions of the american mathematical society Volume 334, Number 2, December 1992 GENERATION

transactions of theamerican mathematical societyVolume 334, Number 2, December 1992

GENERATION AND PROPAGATION OF INTERFACESIN REACTION-DIFFUSION SYSTEMS

XINFU CHEN

Abstract. This paper is concerned with the asymptotic behavior, as e \ 0,of the solution (ue, Ve) of the second initial-boundary value problem of thereaction-diffusion system:

J uÇ-eAu* = \f{uc,ve) = ±[ue(l -ue2)-ve],

\ vf - Ave = ue - yve

where y > 0 is a constant. When v € (-2\/3/9, 2\/3/9), / is bistable inthe sense that the ordinary differential equation ut = f(u, v) has two stablesolutions u = h-(v) and u = h+(v) and one unstable solution u = ho(v),where h-(v), h(¡(v), and h+(v) are the three solutions of the algebraic equa-tion f(u, v) = 0. We show that, when the initial data of v is in the interval(-2%/3/9, 2\/3/9), the solution (ue, ve) of the system tends to a limit (u, v)which is a solution of a free boundary problem, as long as the free boundaryproblem has a unique classical solution. The function « is a "phase" functionin the sense that it coincides with h+(v) in one region Q+ and with h-(v)in another region Í2_ . The common boundary (free boundary or interface) ofthe two regions Í2_ and Í2+ moves with a normal velocity equal to ^(v),where 3^(') is a function that can be calculated. The local (in time) existenceof a unique classical solution to the free boundary problem is also established.Further we show that if initially «(•, 0) - hQ{v(-, 0)) takes both positive andnegative values, then an interface will develop in a short time 0(e\ Ine|) nearthe hypersurface where u(x, 0) - ho{v(x, 0)) = 0.

1. INTRODUCTION

This paper is concerned with the interfacial phenomena in the reaction-diffusion system

(1.1) ut — eAu +-f(u, v),

;i.2) vt = Av + g(u,v)

Received by the editors September 28, 1990.1991 Mathematics Subject Classification. Primary 35B25; Secondary 35B40, 35R35, 35K45,

35K50.Key words and phrases. Reaction-diffusion systems, generation of interface, propagation of

interface.This work was completed when the author was a graduate student at the University of Minnesota.

The author thanks Professor Avner Friedman for his direction of this work and the Alfred P. SloanDoctoral Dissertation Fellowship for its financial support during the 1990-1991 academic year.

© 1992 American Mathematical Society0002-9947/92 $1.00+ $.25 per page

877

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878 XINFU CHEN

with

(1.3) f(u,v) = F(u)-v, F(u) = u(l-u2),(1.4) g(u,v) = u-yv,

where y > 0 is a constant and e > 0 serves as a small parameter.The system ( 1.1 ), ( 1.2) models the propagation of chemical waves in excitable

media where u is a propagator and v is a controller (see Fife and Tyson [21]for a physical description of this system; Ohta, Mimura, and Kobayashi [28]also used this system to describe an activator-inhibitor model).

The assumption that e is small means that the propagator u diffuses quiteslowly while its reaction takes place much faster. When v e (-2\/3/9, 2v/3/9),the algebraic equation f(u, v) = 0 has three solutions

u = h-(v), u = h0(v), and u = h+(v),

where h-(v) < ho(v) < h+(v). In this paper, we are only interested in the casewhen v e (-2-\/3/9, 2v/3/9). In this case, / is bistable in the sense that theordinary differential equation ut = f(u, v) has two stable solutions u = h-(v)and u = h+(v) and one unstable solution u = h0(v). The bistable property of/ and the smallness of e are essential to the so-called interfacial phenomenondescribed below.

Starting with smooth initial data, the diffusion term eAu in (1.1) and thevariation of v from its initial data can be neglected for a short time, so thatequation (1.1) can be approximated by the ordinary differential equation ut =\f(u, v(x, 0)), and therefore u(x, t) tends quickly to either h+(v(x, 0)) orh-(v(x, 0)) according to the sign of u(x, 0) - ho(v(x, 0)). Thus, after a shorttime, the space is partitioned into three regions: a region Q+ where u is almostequal to h+(v), a region Q_ where u is almost equal to h-(v), and a "thin"strip region Qq which links Q_ and Q+ . The region Qn is so thin that itcan be considered as a hypersurface, called interface. We refer to the aboveprocess as the generation of the interface. Subsequently, if x is away from theinterface, the diffusion term eAu can still be neglected, and therefore u(x, t)approximately equals h-(v(x, t)) or h+(v(x, t)) depending on which regionx belongs to, whereas v approximately solves (1.2) with g = g(h-(v), v) inone region and g - g(h+(v), v) in the other region. On the other hand, nearthe interface, the change in u is large, so that eAu is nonnegligible. In fact,eAu will approximately balance the reaction term -\f, and their differenceis a force which will drive the interface to move; this motion is called thepropagation of the interface. The normal velocity of the motion of the interfacewill be determined by the speed of a planar travelling wave solution of equation(1.1).

It is well known that (1.1) has a planar travelling wave solution

u = u(x's~ct ,v\ , xe3?n, seSn~] c3ê", te 31,

where U = U(z, v) and c = ^"(v) form the unique solution of the nonlinear


GENERATION AND PROPAGATION OF INTERFACES 879

eigenvalue problem:' Uzz(z,v) + cUz(z,v) + f(U(z,v),v) = 0 1ze3!x,

(1 5) i X{m^+°oU(z,v) = h+(v),limz__00[/(z, v) = h-(v),

k U(0,v) = h0(v).

(For more detailed discussions of the above eigenvalue problem, see Aronsonand Weinberger [2, 3].)

Fife and Hsiao [19] considered the one-dimensional Cauchy problem forequation (1.1) where v is a known function of x. They proved that startingwith smooth initial data u(x, 0) satisfying u(x, 0) > hç,(v(x)) when x > 0,and u(x, 0) < ho(v(x)) when x < 0, the solution of (1.1) approximates thefunction U((x - ¿;(i))/e, v(Ç(t)) as e —> 0, where U(z, v) is the solution of(1.5) and x = ¿;(i) is a function determined by the motion law

n6) U(t) = T(vm))), t>0,U(0) = o.

When v = 0, equation ( 1.1 ) is known as the Allen-Cahn equation [ 1 ]. Its one-dimensional case has been extensively studied by Bronsard-Kohn [6], Carr-Pego[7, 8], Fusco [23], Fusco-Hale [24], and the references therein. Recently, someresults for the Allen-Cahn equation have been extended to higher dimensions.Here the mean curvature K of the interface takes a role in the propagation ofthe interface. Formal derivation shows that the normal velocity of the inter-face is eK (see, for example, Allen-Cahn [1], Rubinstein-Sternberg-Keller [30],and Fife [18]). Rigorous proofs were recently given by Bronsard-Kohn [5],DeMottoni-Schatzman [14, 15], Evans-Soner-Souganidis [16], and the author[9]. The method in [9] is based on the construction of comparison functionsand is flexible enough to be extended to the system (1.1), (1.2), as we shall doin this paper.

For the system ( 1.1 ), ( 1.2), X.-Y. Chen [ 10] has recently proven the generationof the interface. Concerning the propagation of the interface, he derived an e-dependent free boundary problem, proved its local (in time) existence (for anyfixed e > 0), and then formally showed that the solution of the free boundaryproblem approximates the solution of (1.1), (1.2). The free boundary problemis to find a function ve and a free boundary (interface) P which separates32n into two disjoint regions Qe_(t) and Q%(t) at each time t > 0, such that(1.7)

'vf-Avt = g(h+(vt),vt)xai + g(h-(iie),ve)xa-_, xe3ln, t>0,j/p = T~(ve) + eK£,vc(x,0) = y/(x), xe3?n,

, T£(0) = {xe 3ên\y>(x) = h0(y/(x))}, xe3?n,

where Xa stands for the characteristic function of the set A, and Ip andKe are respectively the normal velocity and the mean curvature of the freeboundary P ; the function ue is defined by ue = h+(ve)xsi'+ + h-(v£)xçi*_ .

The present work is an extension of [ 10]. We shall prove the local existence ofa unique solution to the free boundary problem (1.7) with e = 0, which we shallcall the limit free boundary problem. Then we shall show that the solution of


880 XINFU CHEN

the limit free boundary problem approximates the solution of the system (1.1),(1.2). In order to prove this, we need to establish a theorem on the generationof the interface which is a refined version of the one given by Chen [10].

It is worth mentioning that there is a significant difference for the free bound-ary problem (1.7) between the case e > 0 and the case e = 0. The secondequation in (1.7) is parabolic (in local coordinates) when e > 0, but, whene = 0, it is a Hamilton-Jacobi equation, a fully nonlinear first order PDE whichrequires stronger smoothness assumptions on the function ^(v(x, t)) in orderto ensure a unique classical solution.

Our method for proving the uniqueness of the classical solution of the limitfree boundary problem can be applied also to the free boundary problem (1.7)(for e > 0) to derive its uniqueness (existence was already established in [10]).

The special forms of / and g in (1.3) and (1.4) are introduced only for con-venience. In fact, our method applies also to the case when / = f(u, v, x, t)and g = g(u, v , x, t). The essential assumption is that / is bistable and that/ and g are monotone in v and u respectively.

Our method also applies to the Cauchy problem, i.e., to the initial valueproblem (1.1), (1.2) with Q = 32n .

The extension of our results to the case of an arbitrary time interval remainsopen. Although the Hamilton-Jacobi equations (e.g., the second equation inproblem (1.7) with "V(v(x, t)) being a known function) have been extensivelystudied and many global existence results have been established [4, 11, 17, 29,32] (and the references therein), we cannot get the global asymptotic behaviorof the system (1.1), (1.2) since our argument strongly relies on the regularity ofthe solution of the limit free boundary problem.

The plan of this paper is as follows. In §2 we formally derive the equationsof the limit free boundary problem and state our main results. In §3 we estab-lish the law of the generation of the interface (a weaker version of which hasbeen established in [10]). Next, in §§4 and 5, we shall establish the existence,uniqueness, and regularity of the solution of the limit free boundary problem.Finally in §6 we use a comparison lemma for the parabolic system (1.1), (1.2) toestimate the difference between the solution of the limit free boundary problemand the solution of (1.1), (1.2). The difference is of order 0(e|lne|) so that,as e —► 0, the solution of (1.1), (1.2) tends to the solution of the limit freeboundary problem.

Remark 1.1. After this paper was completed, the author was informed thatHilhorst, Nishiura, and Mimura [26] proved the existence of a unique solutionto the limit free boundary problem (1.7) with e = 0 in the one-dimensionalcase, that Evans, Soner, and Souganidis [16] obtained the global asymptoticbehavior of the solution of the Allen-Cahn equation, and that Giga, Goto, andIshii [25] established the global existence of at least a weak solution to the freeboundary problem (1.7) for both the case e > 0 and the case e = 0.

2. Statement of the main resultsConsider the second initial-boundary value problem:

(2.1) uet=eAue + -f(uE,vE) inQx(0,T),



(2.2) vf = AvE + g(uE, vE) inQx(0,T),(2.3) uE(x, 0) = y>(x) forxeQ,(2.4) vE(x,0) = y/(x) for x efi,(2.5) d„ue = 0 for (x,t)edQx[0,T),(2.6) dnvE = 0 for (x,t)edQx[0,T),

where Q is a smooth (C3) bounded domain in 31" (n > 1), dn is thenormal derivative to dQ, and T is any positive number. For simplicity, weshall assume that the functions / and g are given by (1.3) and (1.4).

For the sake of completeness, we first state a well-known existence and unique-ness result for the system (2.1)-(2.6).

Lemma 2.1. Assume that y> and y/ are in C2(Q) and satisfy the compatibilitycondition

(2.7) dn(p = dny/ = Q VxedQ.Then there exists a unique solution of the system (2A)-(2.6) for all 0 < T < +00.Moreover, there exists a positive constant Co such that for all e > 0,

(2.8) \uE(x,t)\ + \vE(x,t)\0,(2.9) \ve(x, t)-y/(x)\ 0.Proof. The existence of a unique solution follows from standard theory forparabolic systems whereas the estimate (2.8) follows by the invariant regiontheory (see, for example, Smoller [31, Chapter 14]). The estimate (2.9) followsby applying the comparison principle to the functions vE and y/(x) ± Cot forthe equation (2.2). D

Observe that for smooth initial data y/ , the change of ve in a short time issmall, so that equation (2.1) is basically the same as the scalar equation (2.1)with ve replaced by y/. This observation leads to the following theorem.

Theorem 1 (Generation of interface). Assume that

0 such that

(2.10) \yi(x)\

882 XINFU CHEN

Remark 2.1. As mentioned earlier, a weaker version of Theorem 1 was firstproved in [10].

The proof, given in §3, is based on the method developed in [9] and involvesthe construction of supsolutions and subsolutions for equation (2.1).

We shall now formally find the asymptotic limit (u,v) of the solution(uE, vE) of (2.1)—(2.6) as £-»0+, leaving the rigorous proof to §6.

Denote by 3¡T the set 3 x [0, T], where 3 is a set in 32N .For 1 < p <+00, it is convenient to introduce the Sobolev norms

ll/II^^^^Ell^l^W + EH^I^r) V*i = l,2, fc = 0,l," i=l (=0

and2 2-i

11/ II W¡-\2¡T) - ¿2 ZZ \\DlD'xf \\V(3>T) ■;=0 7=0

Assuming that the assertion of Theorem 1 holds for each time t > 0, weconclude that there exists a domain D = {Jo


(v, T) in the time interval t e [0, T0], and the solution satisfies v e W¿¿ ' (Qr0)and TeW^2(P>x[0,T0]).

To prove this theorem, we shall first study in §4 the solution T of the prob-lem (2.18), (2.19) where the function T(v) on the right-hand side of (2.18)is replaced by a given function V e W¿;l(QT). Then in §5 we shall estab-lish the W^'1(Qt) regularity of the solution v of the problem (2.15)-(2.17)where D is a given domain compactly contained in QT with a c1+a'(1+a)/2lateral boundary. Denote by 0,

884 XINFU CHEN

In the sequel, we shall denote by the letter C various positive constantswhich do not depend on e .

3. Generation of interface

Notice that (2.9) and (2.10) imply that there exists a positive constant Txsuch that

(3.1) \vs(x,t)\<2V3 1

a V(x, t)eQTl, £6(0, 1)

In the sequel, we shall always assume that T < Tx so that (3.1) holds, andtherefore the three solutions h-(v), ho(v), and h+(v ) of the algebraic equation/(•, v) = 0 are well defined.

In this section we shall show that in a short time of order 0(e\ In e|) the solu-tion (uE, Ve) of the system (2.1)—(2.6) can be approximated by (w(0,

with ¿; e 3lx and v e [-2\/3/9 + \a, 2\/3/9 - \a] as parameters. To dothis, we shall follow the ideas developed in [9]: First we construct a functionw(Ç, x; b) which is the solution of the above ordinary differential equationwith its right-hand side replaced by a modification / of / ; then we show thatfor some constant M large enough, the function

(3.3) u(x, t) = w(tp(x) - Mt, t/e; y/(x) + Me\\ne\)

is a subsolution to the parabolic equation (2.1), and the function

(3.4) w(x, t) = w(q>(x) + Mt, t/e; y/(x) - A/e|lne|)

is a supersolution.In order to define f(u, v), let p(s) e C[f

ing:(32x ) be a cut-off function satisfy-

(3.5)

p(s) = \ if|s|2,

0


Notice that F'(u) = 1 - 3w2 vanishes only at u = ±-\/3/3 ; it is convenientto introduce a constant a defined by(3.7)

yß \ 2yß 1a = min< \n\ F\T + 1l) = ^r-\a orF\

Lemma 3.1. There exist positive constants e and C such (hat for all e e (0, e]and v e [-2V3/9 + \a, 2\/3/9 - \o], the function f defined in (3.6) satisfies

(3.8)

(3.9)(3.10)

(3.11)

\f(u, v)-f(u, v)\ < Ce|lne| Mue [-Co, C0]:

\fu(U,V)\

886 XINFU CHEN

Differentiating (3.16) with respect to u yields

(3.17) fu{u,V) = Poel IneIu- ho

-/ + Polnel + [1 - Po]fu(u,v).

Inequality (3.9) thus follows by (3.15) and the boundedness of p' and fu .To prove (3.11), notice that fu(u, v) > l/|lne| for any u e [-\/3/3 +

~ä,\n>ß-~ä] and e small enough; it follows that the function (w — Ao)/| lne| —f(u, v) has the same sign as ho - u . Since p'0 also has the same sign as ho-u,the first term on the right-hand side of (3.17) is nonnegative. It then followsfrom (3.17) that

/«(", v) > Polne + [1 - Po]fu(u,v)>1

lne|

i.e., (3.11) holds.Differentiating (3.16) with respect to v , we get

fu(u,v) = p'oK0e| lnel« - ho\aW^\ f

PohpI lnel + [1 - Po]fv(u,v).

Comparing this with equation (3.17), we find that (3.14) and (3.10) hold. Dif-ferentiating f, with respect to v and using (3.15) and the boundedness of p" ,we get

Jvv ~ (e\\ne\)2u-hpI lnel -/ +

1e|lne [terms of order 0(1)] "(-irr)\e|lne|/

Therefore, Lemma 3.1 holds in subcase (i). Similarly, we can treat subcases (ii)and (iii) and therefore establish the assertion of the lemma. D

We now define the function w (Ç, x ; v ) as the solution of the ordinary dif-ferential equation

(3.18)( iï)r(Ç,x;v) = f(w,v), t>0,

where £, e [-Co, Co] and v e [-2y/3/9+^o, 2v/3/9-20;

(2) there exist positive constants To and eo such that if e e (0, eo] andx > To|lne|, then w(i, x; v) satisfies

(3.20) w(Ç,x;v)>h+(v)-2e\lne\ V¿¡ e [h0(v) + 2e|lne|, oo),(3.21) t&(i, t; v)< A_(t;) + 2e|lne| Vf e (-oo, h0(v) - 2e|lne|],

and

(3.22) h-(v) -2e\lne| < t&({, x;v)


(3) there exists a positive constant Cx depending only on eo and To definedin (2) such that if e e (0, e0] and x e [0, t0| lne|], then

(3.23) |t%| < Cxw(/e,(3.24) \Wv\ < Cx(l + wç),(3.25) |%| h+. Consequently, ast / +00, we have

(3 28) |t»(i,T;t;)-M«)l\0 foralli < M«),i- |ti)(i, t;v)-A+(«)I\0 for all í > Ao(«) ■Since there exists a constant a > 0 such that

/(«, u) > a min{u - h0, h+- u} Mu e [h0, h+],

by solving (3.18) we find that

îî)(/zo + 2e|lne|, t; v) > min < ° + , A0 + 2e|lne|eaT >

h^,x;v^>h+-max[2e\lne\,f^e-

It follows thath0 + h+

II) '(h0 + 2e\lne\, y|lne|; v) >

and. fh0 + h+ t0|, . \ , . .. .w I —=— , -y|lne| ; w I > A+ - 2e|lne|

provided that we take To = 2/q and e small enough. Therefore, for all £ >Ao + 2e|lne| and T>To|lne|

"w(^, x;v)> i&(A0 + 2e|lne|, x; v)

>w(f^±,x-^\lne\;v)>h+-2e\\ne\,

i.e., inequality (3.20) holds. Similarly, we can show that (3.21) and (3.22) hold.The second assertion of the lemma thus follows.

To prove the last assertion of the lemma, we first consider the case

|Î-AoO)| -Toe|lne|.


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Since f(u, v) = (u - ho(v))/\lne\ when \u - ho(v)\ < e|lne|, we can solve(3.18) explicitly to get

w(Ç, t; v) = h0(v) + (£- Ao(v))eT/IM Mxe[0, T0|lne|].

Direct differentiation then yields

%(£, x;v) = 0,%,(£, t;í;) = 0,wv(Ç,x;v) = h'p(v)[l-e^nE\],

wvv(Ç,T;v) = hQ'(v)[l-e*/^].Assertion (3) thus holds in this case. Similarly, we can treat the cases |¿; -h-(v)\ < e| Ine| and |¿; - h+(v)\ < e|lne|.

It now remains to consider the case when Ç e B(v), where

B(v) = {ne [-C0, Cp]\\n - h±(v)\ >e|lne| and \n - h0(v)\ > e~T°e|lne|}.

We claim that

(3.29) \f(w(cl,x;v),v)\>e-*°e Mx e [0, T0|lne|], Í 6 B(v).In fact, one can easily verify that when t\ e (hp, h+), the function w(£,, x; v)is monotone increasing in x and £, and that the solution of (3.18) with £ =h+ -e|lne| is given by

w(h+ -e|lne|, x; v) = h+ — e| lne|e_T/|lne|.

Hence, when Ç e [/zo+e~T°e|lne|, A+-e|lne|] and x e [0, T0e|lne|], it followsthat

¿Í < iî)(Ç, x;v) {(í,T;t;) = exp(ln|/(z,t;)|£

Differentiating this expression with respect to Ç and taking absolute values, weget

!%(£> t;u)| = fu{™>v)ù(-§^M,v)f(t,v)

fu(W, V)M,v)

Wç -

P(í,v.M,v)M,v) fy(Ç,v]

CeT° .

by (3.29) and the boundedness of /„ ; inequality (3.23) thus follows.


GENERATION AND PROPAGATION OF INTERFACES

Since iI)v(Ç, x; v) satisfies (wv)r — fu(w , v)wv+fv(w , v) and wv(Ç, 0; v)= 0, we have

wv(£,x;v)= expi/ fu(w(Z, x2 ; v), v)dx2 j fv(w(Ç, xx ; v), v)dxx.

Applying the change of variables (3.30) to the integrals on the right-hand sideyields

tB„(i, x;v)= IJo

(3.32) =f(w,v)j

f(w(Ç, x; v), v) ? , _._ , , ,' fv(w(Ç,xx;v),v)dxxo f(w(Ç, xx;v),v)

" fv(z,v) dz

(3.33) f(w

i P(z,v)fv(z,V)

£|z±v/I/3|„| < C(l +wt), and f(w , v) = %/(£, v) ; inequal-ity (3.25) thus follows.

It remains to prove (3.26). Differentiating the expression in (3.32) with re-spect to v yields

fv(z,v)

(3.35)

f f (z v)wvv = [fu(w, v)wv + fv(w, v)] J") ' [dz

Ji f2(z.v)

fv(w,v). ,,. . r+ JV" 'wv+f(w,v) I

f(w,v) Jçfvv(z,v) 2f2(z,v)P(z,v) " p(z,v)

dz.


890 XINFU CHEN

From (3.33) and (3.34), we know that the first integral on the right-hand sideis bounded by

1 1 \C 1 + +|/(i,t>)| \f(W,V)\J

Proceeding as in (3.33) and (3.34), we can also estimate

I Jv\í P(z,v) dz 7f \fu(z,V)\ 1 + 1\M,v)\ \f(w,v)| + e|A^| + |Vç>|2 + |V


One can also compute

Ut-f(u, Ve) = -Wx - MWç-f(w , Ve)

= -f(w , y/ + Me\lne\) - Mw^

—[f(ii , ¥ + Me\ lne|) + y/ + Me\ lne| - vE]

< -|/-/| + -\vE - y/\ - Mwç - M\lne\

< -Mw( -[M-C- Co t0]| In e| Vf e(0, T0e|lne|],

where we have used the relations wT = /, \f — f\< Ce|lne| (by (3.8)), and\vE - y/\ < CoT0e|lne| (by (2.9)). Hence,

ut -eAu-f(u, vE) < (C + CoXo + CxAo-M)\\ne\ + (CxAo- M)w£ < 0

if M is large enough (recall that wç > 0). Inequality (3.36) thus follows.Note that w(Ç, 0; v) =% implies

u(x, 0) = w(0.

Hence, applying a comparison theorem for the parabolic equation (2.1) to thepair of functions uE and u yields

u(x,t) ^(^(x) - Mxoe\\ne\, To|lne| ; y/(x) + A/e|lne|).

Using the first inequality in (3.22) and the Lipschitz continuity of the functionh- , we get

uE(x, T0e|lne|) > h-(y/(x) + Me|lne|) - 2e|lne|> h-(y/(x)) -(cA/ + 2)e|lne|,

which yields the first inequality in (2.11) if we take M large enough.Substituting (3.20) into (3.38), we obtain

uE(x, T0e|lne|) > h+(y/(x) + Me\\ne\) - 2e|lne| > h+(y/(x)) - (cM + 2)e|lne|

provided h0(y/(x) + Me|lne|) + 2e|lne|.

The last condition can be fulfilled if

ho(y/(x)) + Ce\ lne|for some constant C large enough. This establishes (2.12). Theorem 1 thusfollows. D

Remark 3.1. By carefully checking the proof of Theorem 1, one finds that equa-tion (2.2) is used only to derive the a priori estimate (2.9). Hence, equation (2.2)


892 XINFU CHEN

can be replaced by any equation which, when coupled with equations (2.1) and(2.3)-(2.6), forms a system admitting a solution satisfying (2.9). In particular,Theorem 1 remains true if equation (2.2) is replaced by the equation

ve - pAvE = g(uE, Ve),

where p = p(e) is any constant satisfying 0 < p. < C for some positive constantC independent of e .

Remark 3.2. The condition cp e C2 in Theorem 1 can be replaced by theweaker condition that tp is Lipschitz continuous. In fact, the only modificationneeded in the proof is to replace the function tp in (3.3) by (f which satisfies

E <


Let L < \ dist(P, dQ) be a small fixed positive constant and let Op: T°(L)-» 32x be a C2 function such that

r° = {xeP(L)|(Do(x) = 0},D° n T°(L) = {x e P(L)| 0},

|V0 VxeP(Z).Given V e W^X(QT), we extend it to a W^x function in 32N x(0, T)

and consider the following system of ordinary differential equations:

^ dS^^-mâ^*^^(4.6) ^ß(x, í) = \Q(x, t)\VV(Y(x,í),í).(4.7) T(x,0) = x,

(4.8) ö(x,0)=VO°W|Vd>o(x)| '

where x e T°(L) is considered as a parameter.This system has a unique solution as long as \Q\ remains positive. Assume

that (T, Q) is a solution of (4.5)-(4.8) in some interval 0 < t < ô . Then by(4.6), |£ß| < C\Q\, where C = ||VF||Loo(£V). Since |ß(x, 0)| = 1, it followsthat(4.9) e~Ct < \Q(x, 0| < eCt Mx e T°(L).This a priori bound shows that the system (4.5)-(4.8) admits a unique solutionfor all t e [0, T].

Notice that when x e P, Q(x, 0) is equal to the unit inward normal of Pso that T(x, 0 is independent of the choice of

894 XINFU CHEN

Denote by Fx and F2 the right-hand sides of (4.5) and (4.6) respectively.Differentiating (4.5) and (4.6) with respect to x, we get, for each x e T°(L),

(4.11)

(4.12)

(4.13)

d_(dY/dx\ (dFy/dY dFx/dQ\ fdY/dxdt \dQ/dx) - \dF2/dY dF2/dQ) \dQ/dx

dY_dx

= A(x ,)(dY/dx-A{x,t)\^dQ/dx

(x, 0) = /„ (the unit nxn matrix).

§§d~n~dx x).

Since V e W^¿x and |ß| is strictly positive (by (4.9)), the matrix A isbounded; hence,

fj(*.o,!f(*,o


differential system:

(4.16) *-Y{x,t) = ~V(r,t),

(4.17) jV(x,t) = Vf(Y,t),

(4.18) jtP(x, t) = f(Y, t)P + \P\VV(Y, t) + VVf(X, t),

(4.19) T(x,0) = x,(4.20) ¥(x, 0) = (x, 0),(4.21) P(x, 0) = V(x, 0) = 0, i.e., if and only ifx e P . Therefore, we have

(4.22) Tt = {ye Q\®(y ,0 = 0} = {Y(x, t)\*¥(x, t) = Y(T°, t)}.

One can verify that when x e P (so that ^(x, 0 = 0), the pair (T(x, 0,ß(x, 0), where

ö(x t) = exp(-/07(f(*,T),T)¿T)m*'*'- \P(x,o)\ nx,t),

forms a solution of (4.5)-(4.8); hence, by uniqueness, Y(x, t) = Y(x, t) forall X e P . The assertion of the lemma then follows from (4.22). D

Theorem 4.3. Let V e W¿;X(Q x (0, T)) be given and define t, as in (4.10).Then, for any S e (0, /»), the motion problem admits a unique solution in thetime interval [0, a].Proof. In view of Lemma 4.2, we need only show that {Y(T°, 0}o

896 XINFU CHEN

To prove this, we compute

d („, x dY, ,\ d _ dY _ d ( d „_(ß(X,0—(*,0) = ^ß-^ + ß-^(^

= lÖlVF-fi-ß-^(eF) (by (4.5) and (4.6))

IW|V ôx * ax Viol y iöi V öx

=_öAfö.v=-1ißi-udx\\Q\) 2™dx QIÖI2N

0;

hence,

(4.24) ß(x,0.||(x,0 = ß(x,0).||(x)0) = i^|i Vxel».

On the other hand, the definition of 4> implies that

(4.25) oW Vx e r°(Ä) •

Dividing both sides by | VOol and subtracting the resulting equation from (4.24)yields

(Q_™_)9Y =\u |Vt is also Lipschitz in y andt. Hence,

.

Let a be the minimum value of |0 Mse32x,

G'(s) = 0 Ms e(-oo,-a/2]U [a/2, oo).



SetG(a/2) if x is in the inside of Y(T°(a), t),

- |G"VÔ»|K = [(//, 0) • ̂ f(ri,0) = -G'(0)V

898 XINFU CHEN

In the sequel, we shall denote by S the mapping from v e W^X(QT) to/, the function in (4.31) where Y is the solution of the motion problem withV = -T(v).

Theorem 4.5. Let vx and v2 be two functions in W^X(QT), and set lx =&vx and I2 = S?v2. Then there exists a constant C depending only on M =\\vl\\w^(nT) + \\v2\\fv^{ciT)>such that

(4.36) HZ1 - /^.(PxpMD < Ct\\vx - v2\\wi.o(aw) Mte[0, T(M)].Proof. Denote by (Y'(x, t), Q'(x, t)) the solution of the system (4.5)-(4.8)corresponding to V1 — -"V(vl). Then

^-(Yx-Ydr >-e2:+

= \FX(YX, Qx) - FX(Y2 , Q2)\ + \F2(YX, Qx) - F2(Y2 , Q2)\

< C[\YX - Y2\ + |ß' - ß2| + IVF^y1, 0 - VV2(Y2, oi+ \Vx(Yx,t)-V2(Y2,t)\]

< C[(l + \\Vl\\w,o{QriM]))\Yx - Y2\ + |ß> - ß2| + \\VX - V2\\w^aTm)].

Since Yx = Y2 and ß1 = ß2 at / = 0, we can apply GronwalPs inequality toget

(4.37) |y' - y2| + iß1 - ß2| < ct\\vx - v2\\wuo(anM]).

Denote by Yr> the inverse of Y'. The identity Y'(Yr>(y ,t),t)=y (i =1, 2) implies

(4.38) YX(Y2~' , t) - YX(YX~] , 0 = Yx(Y2~l , t) - Y2(Y2~' , t).

Applying the mean value theorem to the expression on the left-hand side andusing (4.14), we get

|y'(y2"', o - yx(yx~> , oi > 5iy2"' - y1_1i

if t y is small enough; therefore, after using (4.37) to estimate the right-handside of (4.38), we obtain

(4.39) |y'" -y2" |


Recall that the function / is defined implicitly by (4.34). By an argumentsimilar to that which leads to (4.39), we can deduce from (4.40) and (4.41) that

II/1 -/2llc.(n.x[o,í]) < Ct\\Vx - V2\\w^{ClTm) Mte[0, T(M)].

Since V - -^(vl) and the function W is smooth, the assertion of thetheorem follows. D

5. A W2; X ESTIMATE FOR A PARABOLIC EQUATIONAND THE PROOF OF THEOREM 2

Consider the parabolic problem

(5.1) v, - Av = fi(v)xo + fi{v)xiy , xeQ, te(0,T),(5.2) v(x,0) - y/(x), xeQ,(5.3) dnv(x,t) = 0, xedQ, te(0,T),where /!(•) and f2(-) are Lipschitz continuous functions, D = \J0

900 XINFU CHEN

Since /i(-) and f2(-) are Lipschitz continuous, one can directly verify that J?is a contraction mapping under the norm ||| v |||= sup0


\a\a2/2-|¿;'|2/2-C(?-T)1+Q;

it follows that

Y(?,a-k,t-x)< C—^-j-2e-^2+^'^


902

and

XINFU CHEN

//JJO'(a-k T(-


then we can substitute (5.12) and (5.13) into (5.11), obtaining

\vl-v2\\w^{ä,)^Cr' \\vx - v2

/ -ij—-ax + L-\\i -t ||c".»/2(roxr0;i])

Inequality (5.10) thus follows by Gronwall's inequality.It remains to prove (5.13). Using the change of variable Ç = xo(r¡, r) we get

ft r fl'(l>T)N(x,t)= / / g(n,r,x)G(x,t;xp(n,r),x)drdndx,

Jo Jp> Jl2(ri,r)

where

g(n, r, x) = det (^;)°) MviiMl, r), r)) - f2(v2(xp(r,, r), x))].

Denote by m - m(t) the norm ||/1 - /2||z.°°(roX[o,i]) • We can estimate N by

\N(x,t)\x[_L>L]xlo,t)) / sup G(x,t;x0(n,r),x)dndxJo JTVre[-L,L]

904 XINFU CHEN

We can estimate N„ by

|tf»l // / {\J [)Y(n,r + lx-a,t-x)drdndxJjQ'(S,t)J0 2(Í-T)iQ'V.t)i-rn

^\ dr\\ ^7J^Y(r1,r + lx-a,t-x)dr1dxJ-m JJ0'(S.t)2(t-V-m

rmQ'(S,t) 2(t - t)

if lililím~" JjQ'(S,t) 2(Í-T)

+ f" dr if ^!',th(t1,r + lx-a,t-x)dndxJ-m JJo'(S.t) A*-*)


By Theorem 5.1, for every / 6 XTo, l^ljpMrñxro ri) - M *°r some con"stant M independent of / and T0 (if To is small). Thus, by Theorem 4.4,&l = %(%?l) e W^2(Y° x [0, min{T0, T(M)}]) and

ll

906 XINFU CHEN

and that a four-tuple (ü,v , u,v) satisfies

(6.2) üt - eAïï —f(ü, v) > 0 in Qp,(6.3) Vt-Av-g(ü,v)>0 inQT,

(6.4) ut - eAu —f(u ,v)LX,

where Lx e (0, Lo) is a fixed small constant. We can assume that \d(x, t)\ >Li/2 whenever dist(x, P) > Li/2. Taking smaller Lx if necessary, we mayalso assume that dist(0fi, P) > L, for all t e [0, T0]. It follows that

(6.12) dnd(x,t) = 0 M(x,t)edQTo.

Let (U(•, v), T"(v)) be the unique solution of the eigenvalue problem (1.5).



Direct verification shows that when / is given by (1.3), the solution is given byh+(v)-h.(v)U(z,v) = h+(v)

l+exp[j.(z + z0(v))(h+{v)-h-(v))}'

T(v) = -j=(2h0(v)-h+(v)-h-(v)),

where z0(v) e 32x is a constant ensuring the last condition in (1.5). Clearly,the function U satisfies

(6.13) Uz(z,v)>0 Mze32x,(6.14)

\Uz(z,v)\ + \Uzz(z,v)\ + \U(z,v) - h+(v)\ 0,(6.15) \Uz(z,v)\ + \Uzz(z,v)\ + \U(z,v)-h-(v)\

908 XINFU CHEN

In the first case, (2.20) implies that h±(b)+cxa Mb e 2V3 a_ 2y/3 a9 + 2' 9 2 ,«e[o.;

it follows that if

(6.26) h(x,0)> ^e|lne| Vxefi,

. .'rf + Miellne,u(x, 0) = U (-j-1-'- ,y/-2h

then

>h-(y/-2h) > h-(y/) + 2cxh(x, 0)> h-(y/) + 2M0e\lne\ > ue(x, T0e|lne|),

i.e., the second inequality in (6.8) holds.To get the second inequality in (6.8) in case (ii), let a be the positive constant

in (6.14), and set

(6.27) Mx =-1—c0 a

Then in case (ii), we can compute

'd + Mxe\\ne\w(x, 0) = U ,yi-2h

> u(-\lne\, y/-2h\ >h+(y/-2h)-Ae~2^E\

> h+(y/) + 2cxh(x, 0) - Ae2 > h+(y/) + 2M0e\lne\ - Ae2> h+(y/) + M0e\lne\ > ue(x, T0e|lne|),

where we have used the monotonicity of U(-, v) in the first inequality, (6.14)in the second inequality, (6.25) in the third inequality, (6.26) in the fourth in-equality, and Theorem 1 in the last inequality. Therefore, the second inequalityin (6.8) holds. Similarly, we can prove that the first inequality in (6.8) holdsunder the condition (6.26) and the choice of M\ in (6.27).

Finally, we verify the differential inequalities (6.2)-(6.5).First we consider (6.2). Denote by U, Uz , Uv , etc. the functions U,

Uz, Uv , etc., evaluated at ((d + Mxe\\ne\eMt)/e,v-2h). Direct computation



yields

% - eA«-f(ü, v)

= {vAMMf*M'+VA«,-2l,l]}

{ el e e

+ Vvv\V(v - 2h)\2 + VvA(v - 2/z) j

--£{f(V,v-2h)-h),

= l^rldt + MM^Ilnel^' + T(u - 2/z)] j + j-Z7zz(l - \Vd\2)\

- (UzAd + 2VzvVd • V(v - 2h) + eTJvv\V(v - 2h)\2

+ Vv[eA(v - 2/z) - (v - 2h)t]} + -

r r r ^= 7i +12 + h + - ■ewhere in the second equality we have used the equation satisfied by U.

To estimate 7!, notice that

(6.28) dt(x, t) = -T(v(x, 0) Vx e P (= {x e Q\d(x, t) = 0})since the outward normal velocity of Yt is dt and (Y, u, v) is a solution of thelimit free boundary problem. By the mean value theorem and the smoothnessof the function d, equation (6.28) implies

\dt(x, t) + ^(v(x, 0)1 < C\d(x, 01 V(x, 0 e QTo.It follows that

dt + MMxe\ \ne\eMt + T~(v - 2/z)= dt + T(v) + MMxe\ \ne\eMt + [T(v - 2/z) - T(v)]> -C\d\ + MMxe\\ne\eMt - Ch> -C\d + Mxe\ lne|eM'| + (M- C)Mxe\ lne|eMi - Ch.

Therefore if M and h satisfy(6.29) sup\h(x,t)\ -C\d + Mxe\ \ne\eMt\.

It follows that, ^ Júf + M,e|lne|eM'l fd + Mxe\lne\eMt „,\h>-C-J-U,[-J-,v-2hjh

>-Csup(|z|^-alzl)>-C

by (6.14) and (6.15).


910 XINFU CHEN

Next we estimate I2 . Since |Vú?| = 1 when \d\


(6.33) ht-Ah>Ch + C{e + X{\d\

912 XINFU CHEN

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Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania15260

E-mail address: [email protected]


GENERATION AND PROPAGATION OF INTERFACES IN REACTION-DIFFUSION SYSTEMS · 2018-11-16 · transactions of the american mathematical society Volume 334, Number 2, December 1992 GENERATION

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